Problem: Let $C$ be the curve defined by $y^2 = 2x$. We have a change of variables: $\begin{aligned} x &= X_1(u, v) = \dfrac{u}{2} + \dfrac{v}{4} \\ \\ y &= X_2(u, v) = \dfrac{v}{3} \end{aligned}$ What is $C$ under the change of variables? Choose 1 answer: Choose 1 answer: (Choice A) A $v^2 = 9u + 9v$ (Choice B) B $v^2 = 36u + 2v$ (Choice C) C $2v^2 = 18u + 9v$ (Choice D) D $2v^2 = 36u + 2v$
Solution: When applying a change of variables, we substitute the new definition for $x$ and $y$ into the original equation. The original equation: $y^2 = 2x$ Let's substitute $X_1(u, v)$ for $x$ and $X_2(u, v)$ for $y$. $\begin{aligned} \left( \dfrac{v}{3} \right)^2 &= 2 \left( \dfrac{u}{2} + \dfrac{v}{4} \right) \\ \\ \dfrac{v^2}{9} &= u + \dfrac{v}{2} \\ \\ 2v^2 &= 18u + 9v \end{aligned}$ Therefore, under the change of variables, $C$ becomes: $2v^2 = 18u + 9v$